Electron Interactions: A Field Theoretic Generalization of the Gell-Mann-Brueckner Method and a Calculation of Exchange Effects

The method of Gell-Mann and Brueckner for treating electron interactions in a degenerate electron gas is generalized using the Feynman-Dyson techniques of field theory. A Feynman propagator is constructed for the effective interaction between electrons which takes into account the polarizability of the medium of unexcited particles in the Fermi sea. The well-known plasmon excitation appears as a singularity in this propagator. The plasmon is seen to be a correlated, resonant oscillation of the electron density field which is damped by transferring its energy to less correlated, multiple excitations. General expressions for the plasmon dispersion relation and for the plasmon level width are derived in terms of the polarizability of the many body medium. The self-energies of the lowest states of the electron gas are discussed by using the adiabatic theorem. This enables us to derive an exact expression for the ground state energy in terms of the polarizability. Because of the degeneracy of the excited states of the non-interacting system, the adiabatic transforms of these states are not stationary states of the interacting system. However, as the momenta of the excited particles approach the Fermi momenta these states become asymptotically stationary. For states with only a few excited particles present an independent particle model is valid with the result that only the Feynman propagator for the physical one particle state is needed. This propagator, which is corrected for the virtual polarization of the medium by the particle, provides all the information concerning the energies and damping of the single particle states. The second part of the paper is concerned with the detailed calculation of the effects of the interaction on the properties of an electron gas. The lowest order exchange correction to the plasmon energy is computed and found to be small in all cases of physical interest. However, the lowest order contributions to the plasmon damping are seen to modify the observed cut-off for plasmon excitation in electron energy loss experiments in a not negligible way. In applying the formalism to such experiments we also discuss the stopping power of an electron gas and derived the exact lowest order contribution to the single particle damping rate. Using the self-energy method, the correction to the low temperature specific heat of an electron gas is computed exactly to one higher order in r(s) (the interelectron spacing) beyond the calculation of Gell-Mann. It appears that the series in orders of r(s) converges reasonably well only for r(s) 0.8 the specific heat is enhanced from this value. The change in sign appears to be a result of the Pauli Principle. We conclude from these calculations that the procedure of expansion in orders of r(s) gives useful results for values of r(s) < 2. A formal calculation of the third order correction to the correlation energy is also carried out which will give a further clue concerning the convergence of the method if the integrations can be evaluated. For intermediate densities (2 < r(s) < 6) the general perturbation approach may still be valid but a different approximation procedure for treating the polarization effects is needed